Abstract
In this study we establish some identities or estimates for operator norms and the Hausdorff measure of noncompactness of certain operators on spaces |C_{α}|_{k}, which have more recently been introduced in [14]. Further, by applying the Hausdorff measure of noncompactness, we establish the necessary and sufficient conditions for such operators to be compact and so the some well known results are generalized.
Highlights
Absolute Cesaro spacesLet Σxn be an infinite series with partial sum sn. Let (σαn) and (tαn) be the nth Cesaro mean (C, α) of order α with α > −1 of the sequence (sn) and (nan) respectively, e.i., σαn
In a more recent paper Sarıgol [22] has introduced the space |Cα|k for the case α > −1, k ≥ 1 as the set of all series summable by the method |C, α|k, and shown that it is the domain of the matrix T α,k = in the space lk, the space of all k-absolutely convergent series, where tα00,k = 1 and tαnv,k =
If Q is a bounded subset of the metric space X, the Hausdorff measure of noncompactness of Q is defined by χ(Q) = {ε > 0 : Q has a finite ε-net in X}, and χ is called the Hausdorff measure of noncompactness
Summary
Let Σxn be an infinite series with partial sum sn. Let (σαn) and (tαn) be the nth Cesaro mean (C, α) of order α with α > −1 of the sequence (sn) and (nan) respectively, e.i., σαn. The problems of absolute summability factors and comparison of these methods is studied by many authors in [3,4,5,6,7, 15,16,17, 25,26,27,28,29,30] and the important sequence spaces on the matrix domains have been examined by several authors in [1,2, 8,9, 12, 18,19,20]
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